Improved Determination of Long-Time Asymptotics of the Diffusion Spreadability of Two-Phase Media

The diffusion spreadability is a powerful dynamical probe of the microstructure of two-phase heterogeneous media across length scales [Torquato, S., Phys. Rev. E., 104 054102 (2021)].  The spreadability is solely determined by the two-point correlation of the system, which is characterized by the autocovariance function (real space) or the spectral density (Fourier space). Experimentally, it can be directly linked from NMR as well as diffusion MRI measurements. Its short-, intermediate-, and long-time behavior reflects structural features at small, intermediate, and large length scales, respectively.  The long-time asymptotics of the spreadability and the corresponding large-length-scale features are of particular interest, which can be determined by fitting. Here, we demonstrate that this fitting can be improved by incorporating higher-order terms in the long-time expansion.

The large-distance scaling exponent α can be extracted by fitting the spreadability to its asymptotic power law, whose value indicates the hyperuniformity class of the system: stealthy hyperuniform (α→∞), Class I, II, III hyperuniform (α>0), nonhyperuniform (α=0), or antihyperuniform (α<0). We show that the fitting precision can be improved by incorporating higher-order terms in the long-time asymptotic expansion of the spreadability to the asymptotic power law as the fitting function. In addition, by combining the fitted long-time asymptotic expansion of the spreadability with the small-t expansion into a two-point Padé approximant, we can approximate the spreadability with few parameters to a satisfactory precision. The smoothness of the spectral density at the origin can also be deduced from the fittings, which implies the strength of long-range correlations. Our findings can be readily implemented in either numerical or experimental spreadability analysis.